(3-x)^4+(2-x)^4=(5-2x)^4

3 min read Jun 16, 2024

Solving the Equation (3-x)^4 + (2-x)^4 = (5-2x)^4

This equation appears complex, but we can solve it using some algebraic manipulation and a bit of ingenuity. Here's how:

1. Expanding the Expressions

Let's start by expanding the terms using the binomial theorem or by multiplying each term out directly. This will give us a polynomial equation.

(3-x)^4 = 81 - 108x + 54x^2 - 12x^3 + x^4 (2-x)^4 = 16 - 32x + 24x^2 - 8x^3 + x^4 (5-2x)^4 = 625 - 2000x + 2400x^2 - 1280x^3 + 256x^4

2. Substituting and Simplifying

Now, we substitute these expanded expressions back into the original equation:

81 - 108x + 54x^2 - 12x^3 + x^4 + 16 - 32x + 24x^2 - 8x^3 + x^4 = 625 - 2000x + 2400x^2 - 1280x^3 + 256x^4

Combining like terms:

2x^4 - 1280x^3 + 2346x^2 - 2120x + 492 = 0

3. Finding Solutions

This is a fourth-degree polynomial equation. There are no easy algebraic methods to find exact solutions for all four roots. However, we can use numerical methods or graphing calculators to approximate the solutions.

Note: The original equation might have hidden solutions or a different form when manipulated further, making it challenging to find all solutions algebraically.

4. Potential Strategies

Here are some potential approaches to finding solutions:

Factoring: Attempt to factor the polynomial. While difficult, it might reveal some solutions.
Rational Root Theorem: This theorem helps identify potential rational roots.
Numerical Methods: Techniques like the Newton-Raphson method can approximate solutions.
Graphing: Plot the function to visualize the solutions visually.

Conclusion

Solving the equation (3-x)^4 + (2-x)^4 = (5-2x)^4 involves expanding, simplifying, and ultimately resorting to numerical methods or graphical analysis to approximate solutions. The complexity of the equation highlights the need for diverse techniques in solving polynomial equations.